Regression-based Polynomial Chaos Research Articles

Radial Basis Function Surrogates for Uncertainty Quantification and Aerodynamic Shape Optimization under Uncertainties

This paper investigates the adequacy of radial basis function (RBF)-based models as surrogates in uncertainty quantification (UQ) and CFD shape optimization; for the latter, problems with and without uncertainties are considered. In UQ, these are used to support the Monte Carlo, as well as, the non-intrusive, Gauss Quadrature and regression-based polynomial chaos expansion methods. They are applied to the flow around an isolated airfoil and a wing to quantify uncertainties associated with the constants of the γ−R˜eθt transition model and the surface roughness (in the 3D case); it is demonstrated that the use of the RBF-based surrogates leads to an up to 50% reduction in computational cost, compared with the same UQ method that uses CFD computations. In shape optimization under uncertainties, solved by stochastic search methods, RBF-based surrogates are used to compute statistical moments of the objective function. In applications with geometric uncertainties which are modeled through the Karhunen–Loève technique, the use on an RBF-based surrogate reduces the turnaround time of an evolutionary algorithm by orders of magnitude. In this type of applications, RBF networks are also used to perform mesh displacement for the perturbed geometries.

Enhancing the explainability of regression-based polynomial chaos expansion by Shapley additive explanations

Enhancing the explainability of regression-based polynomial chaos expansion by Shapley additive explanations

Dynamic analysis of geared transmission system for wind turbines with mixed aleatory and epistemic uncertainties

This paper deals with the co-existence of mixed aleatory and epistemic uncertainties in a wind turbine geared system for more reliable and robust vibration analyses. To this end, the regression-based polynomial chaos expansion (PCE) is used to track aleatory uncertainties, and the polynomial surrogate approach (PSA) is developed to treat the epistemic uncertainties. This non-intrusive dual-layer framework shares the same collocation pool, which is extracted from the Legendre series. Moreover, the regression technique has been implemented in both layers to enhance calculation efficiency. Numerical validation is carried out to show the effectiveness of the proposed method. New vibration behaviors of the geared transmission system are observed, and the mechanism behind is discussed in detail. The findings of this paper will contribute to the insightful understanding of such wind turbine geared systems under hybrid uncertainties and are beneficial for the condition monitoring.

Multifidelity Modeling by Polynomial Chaos-Based Cokriging to Enable Efficient Model-Based Reliability Analysis of NDT Systems

This work proposes a novel multifidelity metamodeling approach, the polynomial chaos-based Cokriging (PC-Cokriging). The proposed approach is used for fast uncertainty propagation in a reliability analysis of nondestructive testing systems using model-assisted probability of detection (MAPOD). In particular, PC-Cokriging is a multivariate version of polynomial chaos-based Kriging (PC-Kriging), which aims at combining the advantages of the regression-based polynomial chaos expansions and the interpolation-based Kriging metamodeling methods. Following a similar process as Cokriging, the PC-Cokriging advances PC-Kriging by enabling the incorporation of multifidelity physics information. The proposed PC-Cokriging is demonstrated on two analytical functions and three ultrasonic testing MAPOD cases. The results show that PC-Cokriging outperforms the state-of-the-art metamodeling approaches when providing the same number of training points. Specifically, PC-Cokriging reduces the high-fidelity training sample cost of the Kriging and PCE metamodels by over one order of magnitude, and the PC-Kriging and conventional Cokriging multifidelity metamodeling by up to $$50\%$$ to reach the same accuracy level (defined by the root mean squared error being no greater than $$1\%$$ of the standard deviation of the testing points). The accuracy and robustness of the proposed method of the key MAPOD metrics versus various detection thresholds are investigated and satisfactory results are obtained.

Efficient uncertainty quantification of stochastic heat transfer problems by combination of proper orthogonal decomposition and sparse polynomial chaos expansion

To increase the contribution and reliability of computational fluid dynamics efforts in design process of industrial equipments, it is necessary to quantify the effects of uncertainties on the system performance. Due to exponentially increment of the computational cost with number of uncertain variables for uncertainty quantification using classical polynomial chaos expansion methodology, reducing the required number of samples for uncertainty quantification is a real engineering challenge. In this paper, the proper orthogonal decomposition method based on the multifidelity approach is combined with the full and sparse polynomial chaos expansions for efficient uncertainty quantification of complex heat transfer problems with large number of random variables. The conjugate conduction heat transfer in NASA C3X cooled gas turbine blade with geometrical uncertainties and the convective heat transfer in ribbed passage with the stochastic wall heat flux boundary condition are considered as the test cases. Results of uncertainty quantification analysis in both test cases showed that proposed multi-fidelity approaches are able to produce the statistical quantities with much lower computational cost compare to the classical regression-based polynomial chaos method. It is shown that the combination of the proper orthogonal decomposition with the sparse polynomial chaos gives a computational gain at least 2 times greater than combination of the proper orthogonal decomposition with the full polynomial chaos expansion.

Sensitivity analysis based on non-intrusive regression-based polynomial chaos expansion for surgical mesh modelling

The modelling of a system containing implants used in ventral hernia repair and human tissue suffers from many uncertainties. Thus, a probabilistic approach is needed. The goal of this study is to define an efficient numerical method to solve non-linear biomechanical models supporting the surgeon in decisions about ventral hernia repair. The model parameters are subject to substantial variability owing to, e.g., abdominal wall parameter uncertainties. Moreover, the maximum junction force, the quantity of interest which is worthy of scrutiny due to hernia recurrences, is non-smooth. A non-intrusive regression-based polynomial chaos expansion method is employed. The choice of regression points is crucial in such methods, thus we study the influence of this choice on the quantity of interest, and look for an efficient strategy. For this purpose, several aspects are studied : (i) we study the quality of the quantity of interest, i.e. accuracy of the mean and standard deviation, (ii) we perform a global sensitivity analysis using Sobol sensitivity indices. The influence of uncertainties of the chosen variables is presented. This study leads to the definition of an efficient numerical simulation dedicated to our model of implant.

Sparse Linear Regression (SPLINER) Approach for Efficient Multidimensional Uncertainty Quantification of High-Speed Circuits

This paper presents a novel linear regression-based polynomial chaos (PC) approach for the efficient multidimensional uncertainty quantification of general distributed and lumped high-speed circuit networks. The key feature of this paper is the development of a modified Fedorov search algorithm based on the ${D}$ -optimal criterion that expeditiously locates a highly sparse set of nodes within the multidimensional random space where the original network needs to be probed. Specifically, the number of selected nodes is kept equal to the number of unknown PC coefficients of the network response, thereby making this approach substantially more efficient than the conventional linear regression approach which is based on an oversampling methodology. Additionally, due to the ${D}$ -optimal criterion, this approach ensures highly accurate recovery of the PC coefficients. The validity of this paper is established through multiple numerical examples.